Anna Zamansky

NON-DETERMINISTIC SEMANTICS FOR LOGICAL SYSTEMS

The principle of truth-functionality (or compositionality) is a basic principle in many-valued logic in general, and in classical logic in particular. According to this principle, the truth-value of a complex formula is uniquely determined by the truth-values of its subformulas. However, real-world information is inescapably incomplete, uncertain, vague, imprecise or inconsistent, and these phenomena are in an obvious conflict with the principle of truth-functionality. One possible solution to this problem is to relax this principle by borrowing from automata and computability theory the idea of non-deterministic computations, and apply it in evaluations of truth-values of formulas. This leads to the introduction of non-deterministic matrices (Nmatrices) — a natural generalization of ordinary multi-valued matrices, in which the truth-value of a complex formula can be chosen nondeterministically out of some non-empty set of options. There are many natural motivations for introducing non-determinism into the truth-tables of logical connectives.

Ideal Paraconsistent Logics

We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n-valued logics, each one of which is not equivalent to any k-valued logic with k < n.

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core” of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.

A `Natural Logic' inference system using the Lambek calculus

This paper develops an inference system for natural language within the ‘Natural Logic’ paradigm as advocated by van Benthem (1997), Sánchez (1991) and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek-based system we propose extends the system by Fyodorov et~al. (2003), which is based on the Ajdukiewicz/Bar-Hillel (AB) calculus Bar Hillel, (1964). This enables the system to deal with new kinds of inferences, involving relative clauses, non-constituent coordination, and meaning postulates that involve complex expressions. Basing the system on the Lambek calculus leads to problems with non-normalized proof terms, which are treated by using normalization axioms.

Modular Construction of Cut-free Sequent Calculi for Paraconsistent Logics

This paper makes a substantial step towards automatization of Para consistent reasoning by providing a general method for a systematic and modular generation of cut-free calculi for thousands of Para consistent logics known as Logics of Formal (In)consistency. The method relies on the use of non-deterministic semantics for these logics.

Many-valued non-deterministic semantics for first-order logics of formal (in)consistency

A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costas approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costas approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family of first-order LFIs (which includes da Costas original system C1*, as well as thousands of other logics). We show that our semantics is effective and modular, and we use this effectiveness to derive some important properties of logics in this family.

Quantification in non-deterministic multi-valued structures

In this paper the concept of a multi-valued non-deterministic (propositional) matrix, in which non-deterministic computations of truth values are allowed, is extended to languages with quantifiers. We describe the difficulties involved in applying the two main classical approaches to interpreting quantifiers, the objectual and the substitutional, and solve the difficulties in the case of the latter. Then we turn to the two-valued case, and explore the effects in this context of each of the four standard Gentzen-type rules for the classical quantifiers. As an example, a sound and complete two-valued non-deterministic semantics is provided for a family of first-order proof systems.

Cut-Elimination and Quantification in Canonical Systems

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.

How Cognitively Effective is a Visual Notation? On the Inherent Difficulty of Operationalizing the Physics of Notations

The Physics of Notations [9] (PoN) is a design theory presenting nine principles that can be used to evaluate and improve the cognitive effectiveness of a visual notation. The PoN has been used to analyze existing standard visual notations (such as BPMN, UML, etc.), and is commonly used for evaluating newly introduced visual notations and their extensions. However, due to the rather vague and abstract formulation of the PoN’s principles, they have received different interpretations in their operationalization. To address this problem, there have been attempts to formalize the principles, however only a very limited number of principles was covered. This research-in-progress paper aims to better understand the difficulties inherent in operationalizing the PoN, and better separate aspects of PoN, which can potentially be formulated in mathematical terms from those grounded in user-specific considerations.

Serious Games: Is Your User Playing or Hunting?

There is an increasing demand for entertainment applications developed for pets, in particular for dogs and cats. However, play interaction between animals and technological devices still remains an uncharted territory both for animal behavior and entertainment computing scientific communities. While there is a lot of anecdotal evidence of pets playing digital games, the nature of animal-computer play interactions is still not understood. In this paper we report on empirical findings based on observing and analyzing dog-tablet game interactions. Using categories emerging from our data analysis, we construct an ethogram, a “catalogue” of behavioral patterns typical of dog-tablet interactions. Based on our data analysis, we hypothesize that the nature of the observed interactions is that of predatory behavior, in response to stimuli in the form of “prey-like” virtual objects displayed on the screen. Based on our hypothesis, we further propose some questions for future investigation, and raise some issues that need to be addressed by game developers when targeting dogs as their users.